Geodesic
Complexity of deciding the first-order theory of real closed fields
Zapiski Nauchnykh Seminarov POMI, Computational complexity theory. Part 3, Tome 174 (1988), pp. 53-100

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Let a formula with $a$ quantifier alternations be given having atomic subformulas of the kind ($f_j\geqslant0$) with polynomials $f_i$ as in [5]. Deciding algorithm is designed with complexity $(M(kd)^{(O(n))^{5a-2(m+1)}}\cdot d_0^{(m+n)})^{O(1)}$. 
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     author = {D. Yu. Grigor'ev},
     title = {Complexity of deciding the first-order theory of real closed fields},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {53--100},
     publisher = {mathdoc},
     volume = {174},
     year = {1988},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1988_174_a2/}
}
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D. Yu. Grigor'ev. Complexity of deciding the first-order theory of real closed fields. Zapiski Nauchnykh Seminarov POMI, Computational complexity theory. Part 3, Tome 174 (1988), pp. 53-100. http://geodesic.mathdoc.fr/item/ZNSL_1988_174_a2/